Question

Consider the function $f\left(x\right)=co{s}^{-1}\left(\left[2^x\right]\right)+si{n}^{-1}\left(\left[2^x\right]-1\right)$ , then

(where [.] represents the greatest integer part function)

• A. Domain of $f(x)$ is $x\in -\infty ,0]$
• B. Range of $f\left(x\right)$ is singleton
• C. $f\left(x\right)$ is an even function
• D. $f\left(x\right)$ is an odd function

Answer 1

Answer: (B)

For $f\left(x\right)$ to be defined $\left[2^x\right]=0,1$
$\therefore 2^x\in \left[0,2\right)\Rightarrow x\in \left(-\infty ,1\right)$
Range $\left\{co{s}^{-1}\left(0\right)+si{n}^{-1}\left(-1\right),co{s}^{-1}\left(1\right)+si{n}^{-1}\left(0\right)\right\}\equiv \left\{0\right\}$
$\therefore$ $f\left(x\right)=0\forall x\in \left(-\infty ,1\right)$
$So,$ Domain is not symmetric about the origin
$\therefore$ $f\left(x\right)$ is neither even nor odd